Question: Nagrom the Dwarf Queen desires a tunnel through the mountain to connect her two wealthiest cities, Yram and Haras, which lie on either side. In an effort to determine the length of the tunnel, Nagrom first walks $7\text{ km}$ from Yram to a point where she can see both cities. From that point, she measures $29^\circ$ between the cities. Lastly, she walks $6\text{ km}$ to Haras. What is the length of the tunnel? Do not round during your calculations. Round your final answer to the nearest hundredth of a kilometer.
Answer: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $\,\,29^\circ$ $d$ $7\text{ km}$ $6\text{ km}$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=6^2+7^2-2\cdot6\cdot7\cdot\cos(29^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{6^2+7^2-2\cdot6\cdot7\cdot\cos(29^\circ)}\\\\ d&\approx 3.40 \end{aligned}$ The answer The length of the tunnel is $3.40\text{ km}$.